Many types of detection problems involve small signals in the presence of noise. The statistics of the noise are often (though not always) Gaussian.
The automatic system and method for thresholding signals in the presence of gaussian noise (hereafter referred to as the "present invention") may be used for threshold detection in a particle detection tool for vacuum process chambers, for example. The present invention is generally applicable in optical detection of particles, because all current detectors raise their threshold voltage by several decibels above the theoretical value for a given false alarm rate to avoid large increases in their false alarm rates due to signal- and noise-level changes, and thus sacrifice a great deal of sensitivity. This lost sensitivity is costly, since increasing laser power to compensate for it is very expensive and the minimum detectable particle size goes up as the sensitivity goes down. As semiconductor feature sizes shrink, smaller and smaller particles can give rise to killer defects. It is important to continue to reduce the lower detection limit for particle size.
The present invention lends itself to a plethora of other applications as well. The present invention can be used to solve problems involving the counting of threshold-crossings in the presence of noise or the estimation of noise levels, where the probability distribution function of the noise and the bandwidth are known a priori. Some examples include optical communications, ultrasonic ranging, gravity wave detectors, seismometers, and many types of optical measurements, and so on.
Typically, in particle detection systems, an event is detected when the signal voltage crosses a preset threshold voltage. The false alarm rate (FAR), i.e., the rate at which threshold crossings due to noise peaks occur, depends only on the detection noise bandwidth, the ratio .gamma. (gamma) of the threshold voltage V.sub.t to the RMS noise voltage V.sub.n, and the amplitude statistics of the noise. Since for Gaussian noise (as well as several other types) this relationship is a very steep and monotonic one, it is possible in principle to set the threshold so that the FAR is as small as desired.
The problem with this scenario is that the desired FAR may be only one count per hour, day, or year, making verification difficult due to the extremely long counting times required to get decent count statistics. In addition, the true count rate (i.e., counts due to actual events of the type trying to be measured) is generally considerably higher than the false count rate, so that the false counts cannot be counted independently in general.
If the false count rate were a slowly varying function of the threshold, this would not be much of a problem; small shifts in gain, signal power, or threshold voltage would not cause large false alarm rate changes. It turns out that for many types of noise, especially exponential and Gaussian, the FAR is an extremely sensitive function of the threshold voltage. For a single pole rolloff with noise bandwidth B, in pure Gaussian noise, the false alarm rate is given by: ##EQU1## For a threshold ratio .gamma. of 7.18, corresponding to a FAR of 1 count per day with a 1 MHz bandwidth, a 10 percent decrease in .gamma. corresponds to more than a hundredfold increase in the FAR, to 107 per day. Such a shift could easily result from a gradual increase of laser power or a power supply voltage shift in a particular detection system, for example.
There are several ways of fixing this problem. The simplest method, which is to estimate the total drift which is likely to occur, and then raise the threshold high enough that the FAR cannot get too large under any plausible condition, involves sacrificing sensitivity, or, often, paying a large financial penalty to achieve a bigger signal (e.g. buying a more powerful laser).
One somewhat better way is to detect the RMS voltage of the signal, and make the threshold voltage a constant multiple of the RMS value. However, there is no easy way of preventing the desired signals from perturbing the threshold in this case, which will lead to reduced sensitivity and a time-variable detection probability for small events.
U.S. Pat. No. 4,036,057 to Morais appears to teach a type of RMS signal level tracking. Morais adds the "peak" noise level to a fixed threshold in order to keep the threshold some pre-defined amount above the "peak noise level".
In the case of shot noise, where the noise voltage is related to the DC voltage in a simple way, the threshold could be derived from the DC value of the signal. Once again, this suffers from the perturbation of the threshold by the signal, and may be impractical in the presence of large low-frequency noise (as is often the case).